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Register a new userSome inequalities for $k$-colored partition functions
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Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $kgeq2$. This enables us to extend the $k$-colored partition function multiplicatively to a function on $k$-colored partitions, and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.
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In order to provide a unified combinatorial interpretation of congruences modulo $5$ for 2-colored partition functions, Garvan introduced a bicrank statistic in terms of weighted vector partitions. In this paper, we obtain some inequalities between the bicrank counts $M^{*}(r,m,n)$ for $m=2$, $3$ and $4$ via their asymptotic formulas and some $q$-series techniques. These inequalities are parallel to Andrews and Lewis results on the rank and crank counts for ordinary partitions.
In 1917, Hardy and Ramanujan obtained the asymptotic formula for the classical partition function $p(n)$. The classical partition function $p(n)$ has been extensively studied. Recently, Luca and Ralaivaosaona obtained the asymptotic formula for the square-root function. Many mathematicians have paid much attention to congruences on some special colored partition functions. In this paper, we investigate the general colored partition functions. Given positive integers $1=s_1<s_2<dots <s_k$ and $ell_1, ell_2,dots , ell_k$. Let $g(mathbf{s}, mathbf{l}, n)$ be the number of $ell$-colored partitions of $n$ with $ell_i$ of the colors appearing only in multiplies of $s_i (1le ile k)$, where $ell = ell_1+cdots +ell_k$. By using the elementary method we obtain an asymptotic formula for the partition function $g(mathbf{s}, mathbf{l}, n)$ with an explicit error term.
The Tur{a}n inequalities and the higher order Tur{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P{o}lya class. A real sequence ${a_{n}}$ is said to satisfy the Tur{a}n inequalities if for $ngeq 1$, $a_n^2-a_{n-1}a_{n+1}geq 0$. It is said to satisfy the higher order Tur{a}n inequalities if for $ngeq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2geq 0$. A sequence satisfying the Turan inequalities is also called log-concave. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence ${ p(n)}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by Chen that $p(n)$ satisfies the higher order Tur{a}n inequalities for $ngeq 95$. In this paper, we prove this conjecture by using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $ngeq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $mgeq 4$ there exists an integer $N(m)$ such that for $ngeq N(m) $, the polynomials $sum_{k=0}^m {mchoose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.
Let $D=(V,A)$ be an acyclic digraph. For $xin V$ define $e_{_{D}}(x)$ to be the difference of the indegree and the outdegree of $x$. An acyclic ordering of the vertices of $D$ is a one-to-one map $g: V rightarrow [1,|V|] $ that has the property that for all $x,yin V$ if $(x,y)in A$, then $g(x) < g(y)$. We prove that for every acyclic ordering $g$ of $D$ the following inequality holds: [sum_{xin V} e_{_{D}}(x)cdot g(x) ~geq~ frac{1}{2} sum_{xin V}[e_{_{D}}(x)]^2~.] The class of acyclic digraphs for which equality holds is determined as the class of comparbility digraphs of posets of order dimension two.
Let $K_{n}^{r}$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lovasz (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_{n}^{r}$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n geq kr + lfloor frac{k-1}{r+1} rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.
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